Wide-band array antenna

ABSTRACT

The present invention provides a wide-band array antenna using a single real-valued multiplier for each antenna element, which is simple in construction and suitable for WCDMA mobile communication system. A rectangular array antenna is formed by N×M antenna elements. Each antenna element has a frequency dependent gain which is the same for all elements. Each antenna element is connected to a multiplier with a single real-valued coefficient, which is determined by properly selecting number of points on a u-v plane defined for simplifying the design procedure according to the design algorithm of the present invention.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a wide-band array antenna,particularly relates to a wide-band array antenna for improving theperformance of a mobile communication system employing the wide-bandcode division multiple access (WCDMA) transmission scheme.

[0003] 2. Description of the Related Art

[0004] Smart antenna techniques at the base station of a mobilecommunication system can dramatically improve the performance of thesystem by employing spatial filtering in a WCDMA system. Wide-band beamforming with relatively low fractional band-width should be engaged inthese systems.

[0005] The current trend of data transmission in commercial wirelesscommunication systems facilitates the implementation of smart antennatechniques. Major approaches for the designs of smart antenna includeadaptive null steering, phased array and switched beams. The realizationof the first two systems for wide-band applications, such as WCDMArequires a strong implementation cost and complexity. On each branch ofa wide-band array, a finite impulse response (FIR) or an infiniteimpulse response (IIR) filter allows each element to have a phaseresponse that varies with frequency. This compensates from the fact thatlower frequency signal components have less phase shift for a givenpropagation distance, whereas higher frequency signal components havegreater phase shift as they travel the same length.

[0006] Different wide-band beam forming networks have been alreadyproposed in literature. The conventional structure of a wide-band beamformer, that is, several antenna elements each connected to a digitalfilter for time processing, has been employed in all these schemes.

[0007] Conventional wide-band arrays suffer from the implementation oftapped-delay-line temporal processors in the beam forming networks. Insome proposed wide-band array antennas, the number of taps is sometimevery high which complicates the time processing considerably. In arecently proposed wide-band beam former, the resolution of the beampattern at end-fire of the array is improved by rectangular arrangementof a linear array, but the design method requires many antenna elementswhich can only be implemented if micro-strip technology is employed forfabrication.

SUMMARY OF THE INVENTION

[0008] An object of the present invention is to provide a wide-bandarray antenna for sending or receiving the radio frequency signals of amobile communication system, which has a simple construction and has abandwidth compatible with future WCDMA applications.

[0009] To achieve the above object, according to a first aspect of thepresent invention, there is provided a wide-band array antennacomprising N×M antenna elements, and multipliers connected to each saidantenna element, each having a real-valued coefficient, wherein assumingthat said elements are placed at distances of d, and d₂ in directions ofN and M, respectively, the coefficient of each said multiplier isC_(nm), and by defining two variables as v=ωd₁ sin θ/c, and u=ωd₂ cosθ/c, the response of said array antenna can be given as follows:$\begin{matrix}{{H\left( {u,v} \right)} = {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{C_{n\quad m}^{{j{({n - 1})}}v}^{{- {j{({m - 1})}}}u}}}}} & (5)\end{matrix}$

[0010] by appropriately selecting points (u_(0l), v_(0l)) on the u-vplane according to a predetermined angle of beam pattern and the centerfrequency of a predetermined frequency band, the elements b_(l) of anauxiliary vector B=[b₁, b₂, . . . , b_(L)](L<<N×M) can be calculated andthe coefficient C_(nm) of each said multiplier corresponding to eachantenna element can be calculated according to $\begin{matrix}{C_{n\quad m} = {\sum\limits_{l = 1}^{L}{G_{a}^{- 1}b_{l}^{{- {j{({n - 1})}}}v_{o_{l}}}^{{j{({m - 1})}}u_{o_{l}}}}}} & (17)\end{matrix}$

[0011] In the wide-band array antenna of the present invention,preferably said each antenna element has a frequency dependent gainwhich is the same for all elements.

[0012] In the wide-band array antenna of the present invention,preferably the gain of the antenna element has a predetermined value ata predetermined frequency band including the center frequency and at apredetermined angle.

[0013] Preferably, the wide-band array antenna of the present inventionfurther comprises an adder for adding the output signals from saidmultipliers.

[0014] In the wide-band array antenna of the present invention,preferably a signal to be sent is input to said multipliers and theoutput signal of each said multiplier is applied to the correspondingantenna element.

[0015] In the wide-band array antenna of the present invention,preferably said selected points (u_(0l), v_(0l)) on the u-v plane forcomputing the elements of said auxiliary vector B are symmetricallydistributed on the u-v plane.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] These and other objects and features of the present inventionwill become clearer from the following description of the preferredembodiments given with reference to the accompanying drawings, in which:

[0017]FIG. 1 is diagram showing a simplified structure of an embodimentof the wide-band array antenna according to the present invention;

[0018]FIG. 2 shows a 2D u-v plane defined for simplification of thedesign of the beam forming network;

[0019]FIG. 3 is a diagram showing the loci of constant: angle θ on theu-v plane;

[0020]FIG. 4 is a diagram showing the loci of constant: angularfrequency ω on the u-v plane;

[0021]FIG. 5 is a diagram showing the desirable points on the u-v planefor designing the wide-band array antenna;

[0022]FIG. 6 is a diagram showing the configuration of the wide-bandarray antenna used for receiving signals;

[0023]FIG. 7 is diagram showing the configuration of the wide-band arrayantenna used for sending signals;

[0024]FIG. 8 is a diagram showing a two dimensional frequency responseH(u,v) calculated according to the designed coefficients; and

[0025]FIG. 9 is a diagram showing plural directional beam patterns on anangular range including the assumed beam forming angle for differentfrequencies.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0026] Below, preferred embodiments will be described with reference tothe accompanying drawings.

[0027]FIG. 1 shows a simplified structure of a wide-band array antennaaccording to an embodiment of the present invention. As illustrated, thewide-band array antenna of the present embodiment is constituted by N×Mantenna elements E(1,1), . . . ., E(1,M), . . . ., E(N,1), . . . ,E(N,M). Here, it is supposed that each antenna element has a frequencydependant gain which is the same for all elements. The direction of thearriving signal is determined by the azimuth angle θ and the elevationangle β. As in most practical cases, it is assumed that the elevationangles of the incoming signals to the base station antenna array arealmost constant. Here, without loss of generality, the elevation angle βis considered as β=90 degrees. The inter-element spacing for thedirections of N and M are d₁ and d₂, respectively.

[0028] To consider the phase of the arriving signal at the elementE(n,m), the element E(1,1) is considered to be the phase reference pointand the phase of the receiving signal at the reference point istherefore 0. With this assumption, the phase of the signal at theelement E(n,m) is given by the following equation. $\begin{matrix}{{\Phi \left( {n,m} \right)} = {\frac{\omega}{c}\left( {{{d_{1}\left( {n - 1} \right)}\sin \quad \theta} - {{d_{2}\left( {m - 1} \right)}\cos \quad \theta}} \right)}} & (1)\end{matrix}$

[0029] where 1<n<N, 1<m<M. In equation (1), θ is considered as the angleof the arrival (AOA), ω=2πf is the angular frequency and c is thepropagation speed of the signal.

[0030] Note that if the elevation angle β was constant but notnecessarily near 90 degrees, then it is necessary to modify d₁ and d₂ tonew constant values of d₁ sin φ and d₂ sin φ, respectively, which are infact the effective array inter-element distances in an environment withalmost fixed elevation angles.

[0031] In the array antenna of the present embodiment, unlikeconventional wide-band array antennas, it is assumed that each antennaelement is connected to a multiplier with only one single realcoefficient C_(nm). Hence, the response of the array with respect tofrequency and angle can be written as follows: $\begin{matrix}\begin{matrix}{{H_{A}\left( {\omega,\theta} \right)} = \quad {{G_{a}(\omega)}{\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{C_{n\quad m}^{j\frac{\omega}{c}{({{{d_{1}{({n - 1})}}\sin \quad \theta} - {{d_{2}{({m - 1})}}\cos \quad \theta}})}}}}}}} \\{= \quad {{G_{a}(\omega)}{H\left( {\omega,\theta} \right)}}}\end{matrix} & (2)\end{matrix}$

[0032] In equation (2), G_(a)(ω) represents the frequency-dependent gainof the antenna elements. Here, for simplicity, two new variables v and uare defined as follows. $\begin{matrix}{v = {\frac{\omega \quad d_{1}}{c}\sin \quad \theta}} & (3) \\{u = {\frac{\omega \quad d_{2}}{c}\cos \quad \theta}} & (4)\end{matrix}$

[0033] Applying equation (3) and (4) in equation (2) gives the followingequation. $\begin{matrix}{{H\left( {u,v} \right)} = {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{C_{n\quad m}^{{j{({n - 1})}}v}^{{- {j{({m - 1})}}}u}}}}} & (5)\end{matrix}$

[0034] With a minor difference, equation (5) represents a twodimensional frequency response in the u-v plane. The coordinates u andv, as illustrated in FIG. 2, are limited to a range from −πto +π,because for example the variable u can be written as $\begin{matrix}{{u} = {{{{\frac{\omega \quad d_{2}}{c}\cos \quad \theta}} \leq \frac{\omega \quad d_{2}}{c} \leq {\frac{2\pi \quad f}{c}\frac{\lambda_{\min}}{2}}} = {{\frac{2\pi \quad f}{c}\frac{c}{2f_{\max}}} \leq \pi}}} & (6)\end{matrix}$

[0035] Note that for a well-correlated array antenna system, it isrequired that d₁, d₂<λ_(min)/2=1/2f_(max), where λ_(min) and f_(max) arethe minimum wavelength and the corresponding maximum frequency,respectively. Equation (6) is valid for v as well.

[0036] According to equations (3) and (4), it can be written that$\begin{matrix}{\frac{v}{u} = {{\frac{d_{1}}{d_{2}}\tan \quad \theta} = {\tan \quad \varphi}}} & (7)\end{matrix}$

[0037] In the special case of d₁=d₂, θ and φ are equal, otherwise, φ canbe given by the following equation. $\begin{matrix}{\varphi = {\tan^{- 1}\left( {\frac{d_{1}}{d_{2}}\tan \quad \theta} \right)}} & (8)\end{matrix}$

[0038] Furthermore, the following equation can be given as$\begin{matrix}{{\left( \frac{v}{\omega \quad {d_{1}/c}} \right)^{2} + \left( \frac{u}{\omega \quad {d_{2}/c}} \right)^{2}} = 1} & (9)\end{matrix}$

[0039] Equation (9) demonstrates an ellipse with the center at u=v=0 onthe u-v plane. In the special case of d₁=d₂=d, the equation (9) can berewritten as following $\begin{matrix}{{\upsilon^{2} + u^{2}} = \left( \frac{\omega \quad d}{c} \right)^{2}} & (10)\end{matrix}$

[0040] Equation (10) demonstrates circles with radius ωd/c.

[0041] Equations (8) and (9) represent the loci of constant angle andconstant frequency in the u-v plane, respectively.

[0042]FIGS. 3 and 4 are diagrams showing the two loci of constant angleθ and constant angular frequency X according to equations (8) and (9).Plotting the two loci in FIG. 3 and FIG. 4, is helpful for determinationof the angle and frequency characteristics of the wide-band beam formingin the array antenna of the present embodiment.

[0043] Here, assume that an array antenna system is to be designed withθ=θ₀, and the center frequency is ω=ω₀. A demonstrative plot, showingthe location of the desired points on the u-v plane is given in FIG. 5.This location is limited by φ₀=tan⁻¹(d₁ tan θ₀/d₂) and r₁<r<r_(h), wherer₁ and r_(h) can be given as follows, respectively. $\begin{matrix}{{r_{l} = {\frac{\omega_{l}}{d}\overset{\_}{d}}},{r_{h} = {{\frac{\omega_{h}}{c}\overset{\_}{d}\quad {and}\quad \overset{\_}{d}} = \sqrt{{d_{1}^{2}\sin^{2}\theta_{0}} + {d_{2}^{2}\cos^{2}\theta_{0}}}}}} & (11)\end{matrix}$

[0044] The symmetry of the loci with respect to the origin of the u-vplane results real values of the coefficients C_(nm) for the multipliersof each antenna element. In the ideal wide-band system, the ideal valuesof the function H(u,v) can be assigned as follows. $\begin{matrix}{H_{ideal} = \left\{ {\begin{matrix}{G_{a}^{- 1};} \\{0;}\end{matrix}\begin{matrix}{{\varphi_{0} = {\tan^{- 1}\left( {\frac{_{1}}{_{2}}\tan \quad \theta_{0}} \right)}},{r_{l} < {r} < r_{h}}} \\{otherwise}\end{matrix}} \right.} & (12)\end{matrix}$

[0045] For example, if the elements have band pass characteristicsG_(a)(ω) in the frequency interval of ω_(l)<ω<ω_(h), then G_(a) ⁻¹(ω)will have an inverse characteristics, that is, band attenuation in thesame frequency band. This simple modification in the gain values of theu-v plane makes it possible to compensate to the undesired features ofthe antenna elements.

[0046] It is clear that the ideal case is not implementable withpractical algorithms. So in the array antenna system of the presentembodiment, a method for determination of the coefficients C_(nm) isconsidered. Below, an explanation of the method for determination of thecoefficients C_(nm) for multipliers connected to the antenna elementswill be given in detail.

[0047] For the design of the multipliers, instead of controlling allpoints of the u-v plane, which is very difficult to do, L points on thisplane are considered. These L points are symmetrically distributed onthe u-v plane and do not include the origin, thus L considered an eveninteger. Two vectors are defined as follows.

B=[b ₁ , b ₂ , . . . , b _(L)]^(T)  (13)

H ₀ =[H(u ₀ ₁ , v ₀ ₁ ), H(u ₀ ₂ , v ₀ ₂ ), . . . , H(u ₀ _(L) , u ₀_(L) )]^(T)  (14)

[0048] In equations (13) and (14), the superscript ^(T) stands fortranspose. The elements of the vector H₀ have the same values for anytwo pairs (u_(0l), v_(0l)), where l=1, 2, . . . , L, which aresymmetrical with respect to the origin of the u-v plane. In addition,they consider the frequency-dependence of the elements in a way likeequation (12). The vector B is an auxiliary vector and will be computedin the design procedure.

[0049] Here, assume that H(u,v) is expressed by the multiplication oftwo basic polynomials and then the summation of the weighted result asfollows: $\begin{matrix}{{H\left( {u,\upsilon} \right)} = {\sum\limits_{l = 1}^{L}{{b_{l}\left( {\sum\limits_{n = 1}^{N}^{{j{({n - 1})}}{({\upsilon - \upsilon_{0_{l}}})}}} \right)}\left( {\sum\limits_{m = 1}^{M}^{{- {j{({m - 1})}}}{({u - u_{0_{l}}})}}} \right)}}} & (15)\end{matrix}$

[0050] In fact with this form of H(u,v), the problem of directcomputation of N×M coefficients C_(nm) from a complicated system of N×Mequations is simplified to a new problem of solving only L equations,because normally L is select as L<<N×M. The final task of the beamforming scheme in the present embodiment is to find the coefficientsC_(nm) for each multiplier from b_(l).

[0051] By rearranging equation (14), the relationship between b, and thecoefficient C_(nm) can be given as follows: $\begin{matrix}{{H\left( {u,\upsilon} \right)} = {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{\left\{ {\sum\limits_{l = 1}^{L}{b_{l}^{{- {j{({n - 1})}}}\upsilon_{0_{l}}}^{{j{({m - 1})}}u_{0_{l}}}}} \right\} ^{{j{({n - 1})}}\upsilon}{^{- j}\left( {m - 1} \right)}u}}}} & (16)\end{matrix}$

[0052] Comparing with equation (5), also by using equation (2), thecoefficient C_(nm) is given as follows: $\begin{matrix}{C_{nm} = {\sum\limits_{l = 1}^{L}{G_{a}^{- 1}b_{l}^{{- {j{({n - 1})}}}\upsilon_{0_{l}}}^{{j{({m - 1})}}u_{0_{l}}}}}} & (17)\end{matrix}$

[0053] That is, after calculation of the vector B, the coefficientC_(nm) can be found according to equation (17) It should be noted thatG_(a) ⁻¹ is a function of frequency, and hence, varies with the valuesof u_(0l) and v_(0l). The computation of the vector B is not difficultfrom equation (15). With the definition of an L×L matrix A with theelements {a_(kl)}, 1≦k, l≦L as follows: $\begin{matrix}{a_{kl} = {\sum\limits_{n = 1}^{N}{^{{j{({n - 1})}}{({\upsilon_{0_{k}} - \upsilon_{0_{l}}})}}{\sum\limits_{m = 1}^{M}^{{- {j{({m - 1})}}}{({u_{0_{k}} - u_{0_{l}}})}}}}}} & (18)\end{matrix}$

[0054] From equations (13), (14) and (15), the following equation can begiven.

{tilde over (H)} ₀ =A B  (19)

[0055] Thus, the vector B is obtained as follows:

B=A ⁻¹ {tilde over (H)} ₀  (20)

[0056] It is assumed that the matrix A has a nonzero determinant, sothat its inverse exists. Then, the values of the coefficients C_(nm) arecomputed from equation (17) and the design is complete.

[0057]FIG. 6 and FIG. 7 are diagrams showing the wide-band arrayantennas of the present embodiment used for receiving and sendingsignals, respectively. As described above, the array antenna isconstituted by N×M antenna elements E(1,1), . . . , E(1,M), . . . ,E(N,1), . . . , E(N,M). As illustrated in FIG. 6, when the array antennais applied for receiving signals, these antenna elements are connectedto multipliers M(1,1), . . . , M(1,M), . . . , M(N,1), . . . , M(N,M),respectively. Each antenna element has a frequency dependant gain whichis the same for all elements, and each multiplier M(n,m) (1≦n≦N, 1≦m≦M)has a coefficient C_(nm) of a real value obtained according to thedesign procedure described above. The output signals of the multipliersare input to the adder, and a sum So of the input signals is output fromthe adder as the receiving signal of the array antenna.

[0058] For each arriving angle of the incoming signals, a set of N×Mcoefficients C_(nm) is calculated previously when designing the arrayantenna, thus by switching the coefficient sets for the antenna elementssequentially, the signals arriving from all direction around the antennaarray can be received. That is, the sweeping of the direction of thebeam pattern can be realized by switching the sets of coefficient usedfor calculation in each multiplier but not mechanically turning thearray antenna round.

[0059] As illustrated in FIG. 7, when the array antenna if used forsending the signals, the signal to be sent is input to all of themultipliers M(1,1), . . . , M(1,N), . . . , and M(N,M). the signal ismultiplied by the coefficient C_(nm) at each multiplier then sent toeach corresponding antenna element. The signals radiated from theantenna elements interact with each other, producing a sending signalthat is the sum of the individual signals radiated from the antennaelements. Therefore, a desired beam pattern for sending signals to apredetermined direction can be obtained.

[0060] Bellow, an example of a simple and efficient 4×4 rectangulararray antenna will be presented. First, the procedure of designing ofthe beam forming, that is, the determination of the coefficient of themultiplier connected to each antenna element will be described, then thecharacteristics of the array according to the result of simulation willbe shown.

[0061] Here, the angle of the beam former is assumed to be θ₀=−40degrees with the center frequency of ω₀=0.7πc/d, where d=d₁=d₂. Becauseof the limitation of the number of the points on the u-v plane in thisexample, it is assumed that G_(a)=1. First, four pairs of criticalpoints (u_(0l), v_(0l)) are calculated as follows:

P₁: (u₀ ₁ ,v₀ ₁ )=(u₀ ,v ₀)  (21)

P₂: (u₀ ₂ , v₀ ₂ )=(−u₀, −v₀)  (22)

P₃: (u₀ ₃ , v₀ ₃ )=(v₀, −u₀)  (23)

P₄: (u₀ ₄ , v₀ ₄ )=(−v₀, u₀)  (24)

[0062] In equations (21) to (24), variables u₀ and v₀ have been foundfrom equations (3) and (4), respectively. Then, the vector H₀ can beformed as

{tilde over (H)} ₀ =H ₀=[1,1,0,0]^(T)  (25)

[0063] Next, the matrix A is constructed using equation (18) and thevector B is calculated from equation (20). Finally, coefficients C_(nm)for 1≦m, n≦4 are computed from equation (17). Due to the symmetry of theselected points (u_(ol), v_(0l)) in the u-v plane, the values ofcoefficients C_(nm) are all real. This simplifies the computation inpractical situations.

[0064]FIG. 8 shows the actual two dimensional frequency response H(u,v)calculated from equation (5) according to the coefficients C_(nm)obtained in the design procedure described above. Clearly, there are twopeak points at P1 and P2, and two zeros at P3 and P4, respectively. Theimportant result of this pattern is that in a relatively largeneighborhood of the point corresponding to ω=ω₀, almost a constantamplitude of the frequency response is obtained. That is, the designed4×4 rectangular array antenna gives a wide-band performance when it isdesigned for the center frequency ω₀ of the frequency band.

[0065]FIG. 9 demonstrates this fact more clearly. In FIG. 8, multipledirectional beam patterns at an angular range including the assumed beamforming angle θ₀, that is −40 degrees for different frequencies fromω_(l) to ω_(h) are illustrated. The frequency response according to thisfigure is from ω_(l)=0.6πc/d to ω_(h)=0.8πc/d, that is, a fractionalbandwidth of 28.6 percent. Assuming a WCDMA system with the carrierfrequency of about 2.1 GHz for IMT-2000, that is, a wide-band signalwith a center frequency of f₀=2.1 GHz, the inter-element spacing will befound as follows: $\begin{matrix}{d = {{0.7\quad \pi \frac{c}{2\pi \quad f_{0}}} = {0.05\quad m}}} & (26)\end{matrix}$

[0066] In the WCDMA mobile communication system for IMT-2000, the higherand lower frequencies will be f_(h)=2.4 GHz and f₁=1.8 GHz,respectively. This frequency band includes all frequencies assignment ofthe future WCDMA mobile communication system.

[0067] According to the present invention, a new array antenna with awide band width can be constituted by a rectangular array formed by aplurality of simple antenna elements with a simple real-valuedmultiplier connected to each of the antenna element. The coefficient ofeach multiplier can be found according to the design algorithm of thebeam forming network of the present invention.

[0068] Comparing to the previously proposed wide-band beam formers, thewide-band array antenna of the present invention employs lower number ofantenna elements to realize a wide-band array. In the simulation of thewide-band beam former as described above, an array with 4×4=16 elementshaving a frequency independent beam pattern in the desired angle isobtained.

[0069] Also, in the wide-band array antenna of the present invention,there is no delay element in the filters that are connected to eachantenna element. Therefore the rectangular wide-band array antennawithout time processing can be realized.

[0070] In conventional array antennas, since most of the coefficients ofmultipliers connected to the antenna elements are complex valued, thesignal process in the multipliers is complicated due to the calculationwith the complex coefficients. But according to the wide-band arrayantenna of the present invention, the multiplier connected to eachantenna element has a single real coefficient, so the signal processingis simple and fast, also the dynamic range of the coefficients are muchlower than other time processing based methods.

[0071] Note that the present invention is not limited to the aboveembodiments and includes modifications within the scope of the claims.

What is claimed is:
 1. A wide-band array antenna comprising: N×M antennaelements, and multipliers connected to each said antenna element, eachhaving a real-valued coefficient, wherein assuming that said elementsare placed at distances of d₁ and d₂ in the directions of N and M,respectively, the coefficient of each multiplier is C_(nm), and bydefining two variables as v=ωd₁ sin θ/c, and u=ωd₂ cos θ/c, the responseof said array antenna can be given as: $\begin{matrix}{{H\left( {u,\upsilon} \right)} = {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{C_{nm}^{{j{({n - 1})}}\upsilon}^{{- {j{({m - 1})}}}u}}}}} & (5)\end{matrix}$

by appropriately selecting points (u_(0l), v_(0l)) on the u-v planeaccording to a predetermined angle of beam pattern and the centerfrequency of a predetermined frequency band, the elements b, of anauxiliary vector B=[b₁, b₂, . . . , b_(L)](L<<N×M) can be calculated andthe coefficient C_(nm) of each said multiplier corresponding to eachantenna element can be calculated as follows $\begin{matrix}{C_{nm} = {\sum\limits_{l = 1}^{L}{G_{a}^{- 1}b_{l}^{{- {j{({n - 1})}}}\upsilon_{0_{l}}}^{{j{({m - 1})}}u_{0_{l}}}}}} & (17)\end{matrix}$


2. A wide-band array antenna as set forth in claim 1, wherein said eachantenna element has a frequency dependent gain which is the same for allelements.
 3. A wide-band array antenna as set forth in claim 1, whereinthe gain of the antenna element has a predetermined value at apredetermined frequency band including the center frequency and at apredetermined angle.
 4. A wide-band array antenna as set forth in claim1, further comprises an adder for adding the output signals from saidmultipliers.
 5. A wide-band array antenna as set forth in claim 1,wherein a signal to be sent is input to said multipliers and the outputsignal of each said multiplier is applied to the corresponding antennaelement.
 6. A wide-band array antenna as set forth in claim 1, whereinsaid selected points (u_(0l), v_(0l)) on the u-v plane for computing theelements of said auxiliary vector B are symmetrically distributed on theu-v plane.